Nyströmformer: A Nyström-Based Algorithm for Approximating Self-Attention by Xiong, Y., Zeng, Z., Chakraborty, R., Tan, M., Fung, G., Li, Y., & Singh, V. (2021)

(Xiong et al. 2021)


This paper describes a way of applying the Nyström method for approximating matrix multiplication to transformers. More precisely, the approximation is used in the self-attention mechanism’s softmax calculation.

This approximation adresses one of the biggest downside of attention: its computational complexity. The authors claim that their method reduces it from \(O(n^2)\) to \(O(n)\).

The goal of the method is to efficiently approximate the matrix

\[\begin{align*} S = \text{softmax}\left(\dfrac{QK^T}{\sqrt{d_q}}\right) \end{align*}\]

Why the standard Nyström method won’t work

To apply matrix approximation to this problem, one would start by writing \(S\) as

\[ S = \begin{bmatrix} A_S & B_S \newline F_S & C_S \end{bmatrix}\]

The full matrix \(S\) can be approximated using only \(m\) columns and \(m\) rows

\[ \hat{S} = \begin{bmatrix} A_S \newline F_S \end{bmatrix} A^+ \begin{bmatrix} A_S & B_S \end{bmatrix} \]

Linearized self-attention via the Nyström method



  1. . . "Nystr\"omformer: A Nystr\"om-based Algorithm for Approximating Self-attention". Arxiv:2102.03902 [cs]. http://arxiv.org/abs/2102.03902.
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